![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmasso | Structured version Visualization version GIF version |
Description: In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.) |
Ref | Expression |
---|---|
rprmasso.b | ⊢ 𝐵 = (Base‘𝑅) |
rprmasso.p | ⊢ 𝑃 = (RPrime‘𝑅) |
rprmasso.d | ⊢ ∥ = (∥r‘𝑅) |
rprmasso.r | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
rprmasso.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
rprmasso.1 | ⊢ (𝜑 → 𝑋 ∥ 𝑌) |
rprmasso.y | ⊢ (𝜑 → 𝑌 ∥ 𝑋) |
Ref | Expression |
---|---|
rprmasso | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rprmasso.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∥ 𝑌) | |
2 | rprmasso.y | . . . 4 ⊢ (𝜑 → 𝑌 ∥ 𝑋) | |
3 | rprmasso.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2734 | . . . . 5 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
5 | rprmasso.d | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
6 | rprmasso.p | . . . . . 6 ⊢ 𝑃 = (RPrime‘𝑅) | |
7 | rprmasso.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
8 | rprmasso.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
9 | 3, 6, 7, 8 | rprmcl 33525 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
10 | 3, 5 | dvdsrcl 20381 | . . . . . 6 ⊢ (𝑌 ∥ 𝑋 → 𝑌 ∈ 𝐵) |
11 | 2, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
12 | 7 | idomringd 20744 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | 3, 4, 5, 9, 11, 12 | rspsnasso 33395 | . . . 4 ⊢ (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ ((RSpan‘𝑅)‘{𝑌}) = ((RSpan‘𝑅)‘{𝑋}))) |
14 | 1, 2, 13 | mpbi2and 712 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘{𝑌}) = ((RSpan‘𝑅)‘{𝑋})) |
15 | 7 | idomcringd 20743 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) |
16 | 8, 6 | eleqtrdi 2848 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (RPrime‘𝑅)) |
17 | 4, 15, 16 | rsprprmprmidl 33529 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘{𝑋}) ∈ (PrmIdeal‘𝑅)) |
18 | 14, 17 | eqeltrd 2838 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘{𝑌}) ∈ (PrmIdeal‘𝑅)) |
19 | eqid 2734 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
20 | 6, 19, 7, 8 | rprmnz 33527 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑅)) |
21 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑅 ∈ Ring) |
22 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑋 ∈ 𝐵) |
23 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑌 = (0g‘𝑅)) | |
24 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑌 ∥ 𝑋) |
25 | 23, 24 | eqbrtrrd 5171 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → (0g‘𝑅) ∥ 𝑋) |
26 | 3, 5, 19 | dvdsr02 20388 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((0g‘𝑅) ∥ 𝑋 ↔ 𝑋 = (0g‘𝑅))) |
27 | 26 | biimpa 476 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (0g‘𝑅) ∥ 𝑋) → 𝑋 = (0g‘𝑅)) |
28 | 21, 22, 25, 27 | syl21anc 838 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑅)) → 𝑋 = (0g‘𝑅)) |
29 | 20, 28 | mteqand 3030 | . . 3 ⊢ (𝜑 → 𝑌 ≠ (0g‘𝑅)) |
30 | 19, 3, 6, 4, 7, 11, 29 | rsprprmprmidlb 33530 | . 2 ⊢ (𝜑 → (𝑌 ∈ 𝑃 ↔ ((RSpan‘𝑅)‘{𝑌}) ∈ (PrmIdeal‘𝑅))) |
31 | 18, 30 | mpbird 257 | 1 ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {csn 4630 class class class wbr 5147 ‘cfv 6562 Basecbs 17244 0gc0g 17485 Ringcrg 20250 ∥rcdsr 20370 RPrimecrpm 20448 IDomncidom 20709 RSpancrsp 21234 PrmIdealcprmidl 33442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-rprm 20449 df-subrg 20586 df-idom 20712 df-lmod 20876 df-lss 20947 df-lsp 20987 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-rsp 21236 df-prmidl 33443 |
This theorem is referenced by: unitmulrprm 33535 |
Copyright terms: Public domain | W3C validator |